CHAPTER 2BASIC PLASMA EQUATIONS AND EQUILIBRIUM2.1INTRODUCTIONThe plasma medium is complicated in that the charged particles are both affected by external electric and magnetic fields and contribute to them.The resulting self-consistent system is nonlinear and very difficult to analyze.Furthermore,the inter-particle collisions,although also electromagnetic in character,occur on space and time scales that are usually much shorter than those of the applied fields or the fields due to the average motion of the particles.To make progress with such a complicated system,various simplifying approxi-mations are needed.The interparticle collisions are considered independently of the larger scale fields to determine an equilibrium distribution of the charged-particle velocities.The velocity distribution is averaged over velocities to obtain the macro-scopic motion.The macroscopic motion takes place in external applied fields and in the macroscopic fields generated by the average particle motion.These self-consistent fields are nonlinear,but may be linearized in some situations,particularly when dealing with waves in plasmas.The effect of spatial variation of the distri-bution function leads to pressure forces in the macroscopic equations.The collisions manifest themselves in particle generation and loss processes,as an average friction force between different particle species,and in energy exchanges among species.In this chapter,we consider the basic equations that govern the plasma medium,con-centrating attention on the macroscopic system.The complete derivation of these 23Principles of Plasma Discharges and Materials Processing ,by M.A.Lieberman and A.J.Lichtenberg.ISBN 0-471-72001-1Copyright #2005John Wiley &Sons,Inc.equations,from fundamental principles,is beyond the scope of the text.We shall make the equations plausible and,in the easier instances,supply some derivations in appendices.For the reader interested in more rigorous treatment,references to the literature will be given.In Section2.2,we introduce the macroscopicfield equations and the current and voltage.In Section2.3,we introduce the fundamental equation of plasma physics, for the evolution of the particle distribution function,in a form most applicable for weakly ionized plasmas.We then define the macroscopic quantities and indicate how the macroscopic equations are obtained by taking moments of the fundamental equation.References given in the text can be consulted for more details of the aver-aging procedure.Although the macroscopic equations depend on the equilibrium distribution,their form is independent of the equilibrium.To solve the equations for particular problems the equilibrium must be known.In Section2.4,we introduce the equilibrium distribution and obtain some consequences arising from it and from thefield equations.The form of the equilibrium distribution will be shown to be a consequence of the interparticle collisions,in Appendix B.2.2FIELD EQUATIONS,CURRENT,AND VOLTAGEMaxwell’s EquationsThe usual macroscopic form of Maxwell’s equations arerÂE¼Àm0@H@t(2:2:1)rÂH¼e0@E@tþJ(2:2:2)e0rÁE¼r(2:2:3) andmrÁH¼0(2:2:4) where E(r,t)and H(r,t)are the electric and magneticfield vectors and wherem 0¼4pÂ10À7H/m and e0%8:854Â10À12F/m are the permeability and per-mittivity of free space.The sources of thefields,the charge density r(r,t)and the current density J(r,t),are related by the charge continuity equation(Problem2.1):@rþrÁJ¼0(2:2:5) In general,J¼J condþJ polþJ mag24BASIC PLASMA EQUATIONS AND EQUILIBRIUMwhere the conduction current density J cond is due to the motion of the free charges, the polarization current density J pol is due to the motion of bound charges in a dielectric material,and the magnetization current density J mag is due to the magnetic moments in a magnetic material.In a plasma in vacuum,J pol and J mag are zero and J¼J cond.If(2.2.3)is integrated over a volume V,enclosed by a surface S,then we obtain its integral form,Gauss’law:e0þSEÁd A¼q(2:2:6)where q is the total charge inside the volume.Similarly,integrating(2.2.5),we obtaind q d t þþSJÁd A¼0which states that the rate of increase of charge inside V is supplied by the total currentflowing across S into V,that is,that charge is conserved.In(2.2.2),thefirst term on the RHS is the displacement current densityflowing in the vacuum,and the second term is the conduction current density due to the free charges.We can introduce the total current densityJ T¼e0@E@tþJ(2:2:7)and taking the divergence of(2.2.2),we see thatrÁJ T¼0(2:2:8)In one dimension,this reduces to d J T x=d x¼0,such that J T x¼J T x(t),independent of x.Hence,for example,the total currentflowing across a spatially nonuniform one-dimensional discharge is independent of x,as illustrated in Figure2.1.A generalization of this result is Kirchhoff’s current law,which states that the sum of the currents entering a node,where many current-carrying conductors meet,is zero.This is also shown in Figure2.1,where I rf¼I TþI1.If the time variation of the magneticfield is negligible,as is often the case in plasmas,then from Maxwell’s equations rÂE%0.Since the curl of a gradient is zero,this implies that the electricfield can be derived from the gradient of a scalar potential,E¼Àr F(2:2:9)2.2FIELD EQUATIONS,CURRENT,AND VOLTAGE25Integrating (2.2.9)around any closed loop C givesþC E Ád ‘¼ÀþC r F Ád ‘¼ÀþC d F ¼0(2:2:10)Hence,we obtain Kirchhoff’s voltage law ,which states that the sum of the voltages around any loop is zero.This is illustrated in Figure 2.1,for which we obtainV rf ¼V 1þV 2þV 3that is,the source voltage V rf is equal to the sum of the voltages V 1and V 3across the two sheaths and the voltage V 2across the bulk plasma.Note that currents and vol-tages can have positive or negative values;the directions for which their values are designated as positive must be specified,as shown in the figure.If (2.2.9)is substituted in (2.2.3),we obtainr 2F ¼Àre 0(2:2:11)Equation (2.2.11),Poisson’s equation ,is one of the fundamental equations that we shall use.As an example of its application,consider the potential in the center (x ¼0)of two grounded (F ¼0)plates separated by a distance l ¼10cm and con-taining a uniform ion density n i ¼1010cm 23,without the presence of neutralizing electrons.Integrating Poisson’s equationd 2F d x 2¼Àen i eFIGURE 2.1.Kirchhoff’s circuit laws:The total current J T flowing across a nonuniform one-dimensional discharge is independent of x ;the sum of the currents entering a node is zero (I rf ¼I T þI 1);the sum of voltages around a loop is zero (V rf ¼V 1þV 2þV 3).26BASIC PLASMA EQUATIONS AND EQUILIBRIUMusing the boundary conditions that F ¼0at x ¼+l =2and that d F =d x ¼0at x ¼0(by symmetry),we obtainF ¼12en i e 0l 22Àx 2"#The maximum potential in the center is 2.3Â105V,which is impossibly large for a real discharge.Hence,the ions must be mostly neutralized by electrons,leading to a quasi-neutral plasma.Figure 2.2shows a PIC simulation time history over 10210s of (a )the v x –x phase space,(b )the number N of sheets versus time,and (c )the potential F versus x for 100unneutralized ion sheets (with e /M for argon ions).We see the ion acceleration in (a ),the loss of ions in (b ),and the parabolic potential profile in (c );the maximum potential decreases as ions are lost from the system.We consider quasi-neutrality further in Section 2.4.Electric and magnetic fields exert forces on charged particles given by the Lorentz force law :F ¼q (E þv ÂB )(2:2:12)FIGURE 2.2.PIC simulation of ion loss in a plasma containing ions only:(a )v x –x ion phase space,showing the ion acceleration trajectories;(b )number N of ion sheets versus t ,with the steps indicating the loss of a single sheet;(c )the potential F versus x during the first 10210s of ion loss.2.2FIELD EQUATIONS,CURRENT,AND VOLTAGE 2728BASIC PLASMA EQUATIONS AND EQUILIBRIUMwhere v is the particle velocity and B¼m0H is the magnetic induction vector.The charged particles move under the action of the Lorentz force.The moving charges in turn contribute to both r and J in the plasma.If r and J are linearly related to E and B,then thefield equations are linear.As we shall see,this is not generally the case for a plasma.Nevertheless,linearization may be possible in some cases for which the plasma may be considered to have an effective dielectric constant;that is,the “free charges”play the same role as“bound charges”in a dielectric.We consider this further in Chapter4.2.3THE CONSERVATION EQUATIONSBoltzmann’s EquationFor a given species,we introduce a distribution function f(r,v,t)in the six-dimensional phase space(r,v)of particle positions and velocities,with the interpret-ation thatf(r,v,t)d3r d3v¼number of particles inside a six-dimensional phasespace volume d3r d3v at(r,v)at time tThe six coordinates(r,v)are considered to be independent variables.We illus-trate the definition of f and its phase space in one dimension in Figure2.3.As particles drift in phase space or move under the action of macroscopic forces, theyflow into or out of thefixed volume d x d v x.Hence the distribution functionaf should obey a continuity equation which can be derived as follows.InFIGURE2.3.One-dimensional v x–x phase space,illustrating the derivation of the Boltzmann equation and the change in f due to collisions.time d t,f(x,v x,t)d x a x(x,v x,t)d t particlesflow into d x d v x across face1f(x,v xþd v x,t)d x a x(x,v xþd v x,t)d t particlesflow out of d x d v x across face2 f(x,v x,t)d v x v x d t particlesflow into d x d v x across face3f(xþd x,v x,t)d v x v x d t particlesflow out of d x d v x across face4where a x v d v x=d t and v x;d x=d t are theflow velocities in the v x and x directions, respectively.Hencef(x,v x,tþd t)d x d v xÀf(x,v x,t)d x d v x¼½f(x,v x,t)a x(x,v x,t)Àf(x,v xþd v x,t)a x(x,v xþd v x,t) d x d tþ½f(x,v x,t)v xÀf(xþd x,v x,t)v x d v x d tDividing by d x d v x d t,we obtain@f @t ¼À@@x(f v x)À@@v x(fa x)(2:3:1)Noting that v x is independent of x and assuming that the acceleration a x¼F x=m of the particles does not depend on v x,then(2.3.1)can be rewritten as@f @t þv x@f@xþa x@f@v x¼0The three-dimensional generalization,@f@tþvÁr r fþaÁr v f¼0(2:3:2)with r r¼(^x@=@xþ^y@=@yþ^z@=@z)and r v¼(^x@=@v xþ^y@=@v yþ^z@=@v z)is called the collisionless Boltzmann equation or Vlasov equation.In addition toflows into or out of the volume across the faces,particles can “suddenly”appear in or disappear from the volume due to very short time scale interparticle collisions,which are assumed to occur on a timescale shorter than the evolution time of f in(2.3.2).Such collisions can practically instantaneously change the velocity(but not the position)of a particle.Examples of particles sud-denly appearing or disappearing are shown in Figure2.3.We account for this effect,which changes f,by adding a“collision term”to the right-hand side of (2.3.2),thus obtaining the Boltzmann equation:@f @t þvÁr r fþFmÁr v f¼@f@tc(2:3:3)2.3THE CONSERVATION EQUATIONS29The collision term in integral form will be derived in Appendix B.The preceding heuristic derivation of the Boltzmann equation can be made rigorous from various points of view,and the interested reader is referred to texts on plasma theory, such as Holt and Haskel(1965).A kinetic theory of discharges,accounting for non-Maxwellian particle distributions,must be based on solutions of the Boltzmann equation.We give an introduction to this analysis in Chapter18. Macroscopic QuantitiesThe complexity of the dynamical equations is greatly reduced by averaging over the velocity coordinates of the distribution function to obtain equations depending on the spatial coordinates and the time only.The averaged quantities,such as species density,mean velocity,and energy density are called macroscopic quantities,and the equations describing them are the macroscopic conservation equations.To obtain these averaged quantities we take velocity moments of the distribution func-tion,and the equations are obtained from the moments of the Boltzmann equation.The average quantities that we are concerned with are the particle density,n(r,t)¼ðf d3v(2:3:4)the particlefluxG(r,t)¼n u¼ðv f d3v(2:3:5)where u(r,t)is the mean velocity,and the particle kinetic energy per unit volumew¼32pþ12mu2n¼12mðv2f d3v(2:3:6)where p(r,t)is the isotropic pressure,which we define below.In this form,w is sumof the internal energy density32p and theflow energy density12mu2n.Particle ConservationThe lowest moment of the Boltzmann equation is obtained by integrating all terms of(2.3.3)over velocity space.The integration yields the macroscopic continuity equation:@n@tþrÁ(n u)¼GÀL(2:3:7)The collision term in(2.3.3),which yields the right-hand side of(2.3.7),is equal to zero when integrated over velocities,except for collisions that create or destroy 30BASIC PLASMA EQUATIONS AND EQUILIBRIUMparticles,designated as G and L ,respectively (e.g.,ionization,recombination).In fact,(2.3.7)is transparent since it physically describes the conservation of particles.If (2.3.7)is integrated over a volume V bounded by a closed surface S ,then (2.3.7)states that the net number of particles generated per second within V ,either flows across the surface S or increases the number of particles within V .For common low-pressure discharges in the steady state,G is usually due to ioniz-ation by electron–neutral collisions:G ¼n iz n ewhere n iz is the ionization frequency.The volume loss rate L ,usually due to recom-bination,is often negligible.Hencer Á(n u )¼n iz n e (2:3:8)in a typical discharge.However,note that the continuity equation is clearly not sufficient to give the evolution of the density n ,since it involves another quantity,the mean particle velocity u .Momentum ConservationTo obtain an equation for u ,a first moment is formed by multiplying the Boltzmann equation by v and integrating over velocity.The details are complicated and involve evaluation of tensor elements.The calculation can be found in most plasma theory texts,for example,Krall and Trivelpiece (1973).The result is mn @u @t þu Ár ðÞu !¼qn E þu ÂB ðÞÀr ÁP þf c (2:3:9)The left-hand side is the species mass density times the convective derivative of the mean velocity,representing the mass density times the acceleration.The convective derivative has two terms:the first term @u =@t represents an acceleration due to an explicitly time-varying u ;the second “inertial”term (u Ár )u represents an acceleration even for a steady fluid flow (@=@t ;0)having a spatially varying u .For example,if u ¼^xu x (x )increases along x ,then the fluid is accelerating along x (Problem 2.4).This second term is nonlinear in u and can often be neglected in discharge analysis.The mass times acceleration is acted upon,on the right-hand side,by the body forces,with the first term being the electric and magnetic force densities.The second term is the force density due to the divergence of the pressure tensor,which arises due to the integration over velocitiesP ij ¼mn k v i Àu ðÞv j Àu ÀÁl v (2:3:10)2.3THE CONSERVATION EQUATIONS 31where the subscripts i,j give the component directions and kÁl v denotes the velocity average of the bracketed quantity over f.ÃFor weakly ionized plasmas it is almost never used in this form,but rather an isotropic version is employed:P¼p000p000p@1A(2:3:11)such thatrÁP¼r p(2:3:12) a pressure gradient,withp¼13mn k(vÀu)2l v(2:3:13)being the scalar pressure.Physically,the pressure gradient force density arises as illustrated in Figure2.4,which shows a small volume acted upon by a pressure that is an increasing function of x.The net force on this volume is p(x)d AÀp(xþd x)d A and the volume is d A d x.Hence the force per unit volume isÀ@p=@x.The third term on the right in(2.3.9)represents the time rate of momentum trans-fer per unit volume due to collisions with other species.For electrons or positive ions the most important transfer is often due to collisions with neutrals.The transfer is usually approximated by a Krook collision operatorf j c¼ÀXbmn n m b(uÀu b):Àm(uÀu G)Gþm(uÀu L)L(2:3:14)where the summation is over all other species,u b is the mean velocity of species b, n m b is the momentum transfer frequency for collisions with species b,and u G and u L are the mean velocities of newly created and lost particles.Generally j u G j(j u j for pair creation by ionization,and u L%u for recombination or charge transfer lossprocesses.We discuss the Krook form of the collision operator further in Chapter 18.The last two terms in(2.3.14)are generally small and give the momentum trans-fer due to the creation or destruction of particles.For example,if ions are created at rest,then they exert a drag force on the moving ionfluid because they act to lower the averagefluid velocity.A common form of the average force(momentum conservation)equation is obtained from(2.3.9)neglecting the magnetic forces and taking u b¼0in theÃWe assume f is normalized so that k f lv ¼1.32BASIC PLASMA EQUATIONS AND EQUILIBRIUMKrook collision term for collisions with one neutral species.The result is mn @u @t þu Ár u !¼qn E Àr p Àmn n m u (2:3:15)where only the acceleration (@u =@t ),inertial (u Ár u ),electric field,pressure gradi-ent,and collision terms appear.For slow time variation,the acceleration term can be neglected.For high pressures,the inertial term is small compared to the collision term and can also be dropped.Equations (2.3.7)and (2.3.9)together still do not form a closed set,since the pressure tensor P (or scalar pressure p )is not determined.The usual procedure to close the equations is to use a thermodynamic equation of state to relate p to n .The isothermal relation for an equilibrium Maxwellian distribution isp ¼nkT(2:3:16)so thatr p ¼kT r n (2:3:17)where T is the temperature in kelvin and k is Boltzmann’s constant (k ¼1.381Â10223J /K).This holds for slow time variations,where temperatures are allowed to equilibrate.In this case,the fluid can exchange energy with its sur-roundings,and we also require an energy conservation equation (see below)to deter-mine p and T .For a room temperature (297K)neutral gas having density n g and pressure p ,(2.3.16)yieldsn g (cm À3)%3:250Â1016p (Torr)(2:3:18)p FIGURE 2.4.The force density due to the pressure gradient.2.3THE CONSERVATION EQUATIONS 33Alternatively,the adiabatic equation of state isp¼Cn g(2:3:19) such thatr p p ¼gr nn(2:3:20)where g is the ratio of specific heat at constant pressure to that at constant volume.The specific heats are defined in Section7.2;g¼5/3for a perfect gas; for one-dimensional adiabatic motion,g¼3.The adiabatic relation holds for fast time variations,such as in waves,when thefluid does not exchange energy with its surroundings;hence an energy conservation equation is not required. For almost all applications to discharge analysis,we use the isothermal equation of state.Energy ConservationThe energy conservation equation is obtained by multiplying the Boltzmannequation by12m v2and integrating over velocity.The integration and some othermanipulation yield@ @t32pþrÁ32p uðÞþp rÁuþrÁq¼@@t32pc(2:3:21)Here32p is the thermal energy density(J/m3),32p u is the macroscopic thermal energyflux(W/m2),representing theflow of the thermal energy density at thefluid velocityu,p rÁu(W/m3)gives the heating or cooling of thefluid due to compression orexpansion of its volume(Problem2.5),q is the heatflow vector(W/m2),whichgives the microscopic thermal energyflux,and the collisional term includes all col-lisional processes that change the thermal energy density.These include ionization,excitation,elastic scattering,and frictional(ohmic)heating.The equation is usuallyclosed by setting q¼0or by letting q¼Àk T r T,where k T is the thermal conduc-tivity.For most steady-state discharges the macroscopic thermal energyflux isbalanced against the collisional processes,giving the simpler equationrÁ32p u¼@32pc(2:3:22)Equation(2.3.22),together with the continuity equation(2.3.8),will often prove suf-ficient for our analysis.34BASIC PLASMA EQUATIONS AND EQUILIBRIUMSummarySummarizing our results for the macroscopic equations describing the electron and ionfluids,we have in their most usually used forms the continuity equationrÁ(n u)¼n iz n e(2:3:8) the force equation,mn @u@tþuÁr u!¼qn EÀr pÀmn n m u(2:3:15)the isothermal equation of statep¼nkT(2:3:16) and the energy-conservation equationrÁ32p u¼@@t32pc(2:3:22)These equations hold for each charged species,with the total charges and currents summed in Maxwell’s equations.For example,with electrons and one positive ion species with charge Ze,we haver¼e Zn iÀn eðÞ(2:3:23)J¼e Zn i u iÀn e u eðÞ(2:3:24)These equations are still very difficult to solve without simplifications.They consist of18unknown quantities n i,n e,p i,p e,T i,T e,u i,u e,E,and B,with the vectors each counting for three.Various simplifications used to make the solutions to the equations tractable will be employed as the individual problems allow.2.4EQUILIBRIUM PROPERTIESElectrons are generally in near-thermal equilibrium at temperature T e in discharges, whereas positive ions are almost never in thermal equilibrium.Neutral gas mol-ecules may or may not be in thermal equilibrium,depending on the generation and loss processes.For a single species in thermal equilibrium with itself(e.g.,elec-trons),in the absence of time variation,spatial gradients,and accelerations,the2.4EQUILIBRIUM PROPERTIES35Boltzmann equation(2.3.3)reduces to@f @tc¼0(2:4:1)where the subscript c here represents the collisions of a particle species with itself. We show in Appendix B that the solution of(2.4.1)has a Gaussian speed distribution of the formf(v)¼C eÀj2m v2(2:4:2) The two constants C and j can be obtained by using the thermodynamic relationw¼12mn k v2l v¼32nkT(2:4:3)that is,that the average energy of a particle is12kT per translational degree offreedom,and by using a suitable normalization of the distribution.Normalizing f(v)to n,we obtainCð2p0d fðpsin u d uð1expÀj2m v2ÀÁv2d v¼n(2:4:4)and using(2.4.3),we obtain1 2mCð2pd fðpsin u d uð1expÀj2m v2ÀÁv4d v¼32nkT(2:4:5)where we have written the integrals over velocity space in spherical coordinates.The angle integrals yield the factor4p.The v integrals are evaluated using the relationÃð10eÀu2u2i d u¼(2iÀ1)!!2ffiffiffiffipp,where i is an integer!1:(2:4:6)Solving for C and j we havef(v)¼nm2p kT3=2expÀm v22kT(2:4:7)which is the Maxwellian distribution.Ã!!denotes the double factorial function;for example,7!!¼7Â5Â3Â1. 36BASIC PLASMA EQUATIONS AND EQUILIBRIUMSimilarly,other averages can be performed.The average speed vis given by v ¼m =2p kT ðÞ3=2ð10v exp Àv 22v 2th !4p v 2d v (2:4:8)where v th ¼(kT =m )1=2is the thermal velocity.We obtainv ¼8kT p m 1=2(2:4:9)The directed flux G z in (say)the þz direction is given by n k v z l v ,where the average is taken over v z .0only.Writing v z ¼v cos u we have in spherical coordinatesG z ¼n m 2p kT 3=2ð2p 0d f ðp =20sin u d u ð10v cos u exp Àv 22v 2th v 2d v Evaluating the integrals,we findG z ¼14n v (2:4:10)G z is the number of particles per square meter per second crossing the z ¼0surfacein the positive direction.Similarly,the average energy flux S z ¼n k 1m v 2v z l v in theþz direction can be found:S z ¼2kT G z .We see that the average kinetic energy W per particle crossing z ¼0in the positive direction isW ¼2kT (2:4:11)It is sometimes convenient to define the distribution in terms of other variables.For example,we can define a distribution of energies W ¼12m v 2by4p g W ðÞd W ¼4p f v ðÞv 2d vEvaluating d v =d W ,we see that g and f are related byg W ðÞ¼v (W )f ½v (W ) m (2:4:12)where v (W )¼(2W =m )1=2.Boltzmann’s RelationA very important relation can be obtained for the density of electrons in thermal equilibrium at varying positions in a plasma under the action of a spatially varying 2.4EQUILIBRIUM PROPERTIES 3738BASIC PLASMA EQUATIONS AND EQUILIBRIUMpotential.In the absence of electron drifts(u e;0),the inertial,magnetic,and fric-tional forces are zero,and the electron force balance is,from(2.3.15)with@=@t;0,en e Eþr p e¼0(2:4:13) Setting E¼Àr F and assuming p e¼n e kT e,(2.4.13)becomesÀen e r FþkT e r n e¼0or,rearranging,r(e FÀkT e ln n e)¼0(2:4:14) Integrating,we havee FÀkT e ln n e¼constorn e(r)¼n0e e F(r)=kT e(2:4:15)which is Boltzmann’s relation for electrons.We see that electrons are“attracted”to regions of positive potential.We shall generally write Boltzmann’s relation in more convenient unitsn e¼n0e F=T e(2:4:16)where T e is now expressed in volts,as is F.For positive ions in thermal equilibrium at temperature T i,a similar analysis shows thatn i¼n0eÀF=T i(2:4:17) Hence positive ions in thermal equilibrium are“repelled”from regions of positive potential.However,positive ions are almost never in thermal equilibrium in low-pressure discharges because the ion drift velocity u i is large,leading to inertial or frictional forces in(2.3.15)that are comparable to the electricfield or pressure gra-dient forces.Debye LengthThe characteristic length scale in a plasma is the electron Debye length l De.As we will show,the Debye length is the distance scale over which significant charge densities can spontaneously exist.For example,low-voltage(undriven)sheaths are typically a few Debye lengths wide.To determine the Debye length,let us intro-duce a sheet of negative charge having surface charge density r S,0C/m2into an。
